In the art of well-logging, physical models are generally applied to represent the electrical behavior of the earth formation. From the well-log results and the physical model, the content of a component of the earth formation is then determined.
A known method of determining parameters of such physical model is disclosed in "Electrical conductivities in oil-bearing shaly sands", Waxman M. H., and Smits L. J. M., SPE paper 1863-A presented at the 42nd Annual Fall Meeting, Houston, Oct. 1-4, 1967. This paper discloses a method of determining a parameter of a physical model representing the electrical behavior of an earth formation, consisting of defining said model by a relationship between the conductivity of the formation, a plurality of physical variables of the formation and said parameter, selecting a sample which is representative for said formation and measuring the electrical conductivity of the sample for various magnitudes of the physical variables, and determining said parameter by applying the selected relationship to the measured conductivities of the sample.
In this known method the model, which is generally referred to as the Waxman-Smits model, is defined by the relationship: ##EQU1## wherein; C.sub.t =the conductivity of the partially brine saturated formation represented by the sample
C.sub.w =the conductivity of brine present in the formation; PA1 S.sub.w =the water saturation in the pore space (0 . . . 1), which equals 1-=S.sub.0 where S.sub.0 denotes the hydrocarbon saturation; PA1 B=the equivalent conductance of sodium clay-exchange cations as a function of C.sub.w and temperature; PA1 Q.sub.v =the cation exchange capacity per unit pore volume; PA1 G*=a formation factor of the formation represented by the sample; PA1 G* is represented as: EQU G*=.phi..sup.-m* S.sub.w.sup.-n* PA1 .phi.=the pore space in the formation; PA1 m*=a parameter to be determined, in the form of the cementation exponent; PA1 n*=a parameter to be determined, in the form of the saturation exponent. PA1 by using conductivity measurements in a non-hydrocarbon bearing zone in the earth formation or with laboratory conductivity measurements on a plurality of fully brine saturated samples; can be determined from the relation between log(C.sub.t.sup.-1) and log(.phi.) or log(C.sub.w .multidot.C.sub.t.sup.-1) and log(.phi.), respectively; and PA1 by using laboratory conductivity measurements on a plurality of partially brine saturated samples n* can be determined from the relation between ##EQU2## and log(S.sub.w). PA1 defining said model by a relationship between the electrical properties of the composition, a plurality of physical variables of the composition and said parameter; PA1 measuring an electrical property of the composition; PA1 selecting a sample which is representative for said composition and measuring an electrical property of the sample for various magnitudes of at least one of said physical variables; PA1 selecting an incoherence function defining a difference between the measured electrical properties of the composition and the sample, and the electrical properties of the composition and the sample as determined from said relationship, said incoherence function being such that independent measurements are weighted in dependence of their accuracy; and PA1 determining said parameter by minimization said incoherence function.
wherein
The parameters m* and n* characterize the response of the conductivity of the earth formation subject to changes in physical variables such as .phi. and S.sub.w. In the known method these parameters are determined in a mutually independent way, namely:
The results achieved with this known method are not always sufficiently accurate, probably because the parameters are determined in a non-optimal manner.
It is an objective of the invention to provide an improved method of determining a parameter of a physical model representing the electrical behavior of a composition.